[PPT] - ligand binding energy James C. (JC) Gumbart Georgia Institute of PowerPoint Presentation

SLIDE 1

ligand binding energy

Accurate calculation of

James C. (JC) Gumbart

Georgia Institute of Technology, Atlanta

Computational Biophysics Workshop | DICP | July 11 2018

Chris Chipot

U. Illinois, Urbana & CNRS, U. Lorraine France

SLIDE 2

I. What is an absolute binding energy?
II. Using restraints to reduce the sampling problem
III. Calculating the requisite PMFs
IV. Comparing geometric with alchemical approach
V. Illustration with barstar-barnase binding

Outline

http://phdcomics.com/

SLIDE 3

Challenge: Absolute binding free energies

protein + ligand protein : ligand

Keq

∆G0 = −kT ln(KeqC◦)

C◦ = 1/1661˚ A

3

N ligands

pi ∝ Zi

1, 2,…,N refer to each ligand 1 is bound (site) in num., unbound (bulk) in denom. Energy does not depend on position of ligand when unbound (bulk is isotropic), so can pick out a specific point x1* and hold it there

C° is the standard concentration of 1 M →

binding free energies are concentration dependent!

SLIDE 4

Illustration using Abl SH3 domain

Chosen ligand: APSYSPPPPP (flexible!) designed to bind with high affinity peptide, so doesn’t require novel parametrization = -7.94 kcal/mol (exp)

MM/PBSA estimate: -2.6 kcal/mol !

Hou, T. et al. PLoS Comput. Biol. 2006, 2, 0046-0055 Pisabarro, M. T.; Serrano, L. Biochemistry 1996, 35, 10634-10640

A well known and conserved domain of Abl kinase

SLIDE 5

geometrical route

Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA, 2005, 102, 6825-6830

How to get Keq and ΔG?

Forcibly separate the ligand from the protein and calculate a PMF

alchemical route

Make the ligand vanish from the binding site and from bulk water Both approaches suffer major sampling deficiencies when used on their own!!!

SLIDE 6

Overcoming sampling issues with restraints

L1 L2 L3 P1 P2 P3

Design set of restraints to reduce conformational

space needed to be sampled

Contributions of each restraint to free energy need to

be rigorously computed Remember! Biasing is okay as long as we can unbias

Bound state RMSD restrained Free state RMSD restrained Assorted spatial/rotational restraints

SLIDE 7

Overcoming sampling issues with restraints

From: Deng and Roux. (2009)

J. Phys. Chem. 113: 2234-2246.

Schematic of process

Conformational Orientational Axial Bound state - turn

ff/on

restraints Free state - turn on/

ff

restraints (un)binding

SLIDE 8

L3

Φ Ψ

ϕ

θ Θ

r

L1 L2 L3 P1 P2 P3 L1 L2 x1

*

Keq = Z

site

d1 Z dx e−βU Z

site

d1 Z dx e−β(U+uc) × Z

site

d1 Z dx e−β(U+uc) Z

site

d1 Z dx e−β(U+uc+uΘ) × Z

site

d1 Z dx e−β(U+uc+uo) Z

site

d1 Z dx e−β(U+uc+uo+uθ) × Z

site

d1 Z dx e−β(U+uc+uo+uθ) Z

site

d1 Z dx e−β(U+uc+uo+uθ+uϕ) × Z

site

d1 Z dx e−β(U+uc+uo+up) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uo) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) Woo, H. J.; Roux, B. (2005) Proc. Natl. Acad. Sci. USA, 102:6825-6830. Yu, Y. B. et al. (2001) Biophys. J., 81:1632-1642.

Binding free energy (geometrical route)

Maffeo, C., Luan, B., Aksimentiev, A. (2012) Nucl. Acids Res. 40:3812-3821. × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

SLIDE 9

How to evaluate all of these integrals?

Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ) Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ+uΨ)

e+β∆Gsite

Ψ

= Z dΨe−β[wsite(Ψ)] Z dΨe−β[wsite(Ψ)+uΨ]

ratio of integrals can be related to a free energy

Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ) Z

site

d1 Z dx (e−βuΨ)e−β(U+uc+uΘ+uΦ) = 1 e−β∆Gsite

Ψ

= e+β∆Gsite

Ψ

Potential of mean force, wsite(ѱ), encapsulates all degrees of freedom = In practice, one determines the PMFs successively and then integrates them as prescribed above

SLIDE 10

Many PMFs are very straightforward

θ θ Φ Φ

two windows used for ABF, 1 ns each

× Z

site

d1 Z dx e−β(U+uc+uo) Z

site

d1 Z dx e−β(U+uc+uo+uθ)

PMFs sampling counts

SLIDE 11

Separation PMF from umbrella sampling

37 windows used, spaced 0.5 - 1 Å apart

histograms are overlapping

× Z

site

d1 Z dx e−β(U+uc+uo+up) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uo)

PMF was already converged within ~20 ns

1 ns/window 0.75 ns 0.5 ns 0.25 ns

−kT ln r2

entropic decay despite no interactions

SLIDE 12

Replica-exchange umbrella sampling (REUS)

helps to circumvent limitations in US by exchanging coordinates

periodically between different windows

exchanges accepted with some probability: min(1, e−∆E/kT )

where ∆E = (wi(ξj) − wi(ξi)) + (wj(ξi) − wj(ξj))

(swapped) (original) (swapped) (original)

See tutorial Methods for Calculating Potentials of Mean Force

SLIDE 13

What you get in the end (a big mess!)

SLIDE 14

Back to the Abl kinase story...

∆Gbulk

c

= 5.43 kcal/mol ∆Gsite

c

= 3.52 kcal/mol

∆Gsep

r

= −14.47 kcal/mol

∆Gbulk

= 5.77 kcal/mol

∆Gsite

= 0.71 kcal/mol

∆Gsite

a

= 0.20 kcal/mol

∆Go = (∆Gbulk

c

− ∆Gsite

c

) + (∆Gbulk

− ∆Gsite
)

+∆Gsep

r

− ∆Gsite

a

= −7.7 kcal/mol

Keq = Z

site

d1 Z dx e−βU Z

site

d1 Z dx e−β(U+uc) × Z

site

d1 Z dx e−β(U+uc) Z

site

d1 Z dx e−β(U+uc+uΘ) × Z

site

d1 Z dx e−β(U+uc+uo) Z

site

d1 Z dx e−β(U+uc+uo+uθ) × Z

site

d1 Z dx e−β(U+uc+uo+uθ) Z

site

d1 Z dx e−β(U+uc+uo+uθ+uϕ) × Z

site

d1 Z dx e−β(U+uc+uo+up) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uo) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

~30 ns 6 ns 4 ns 37 ns 30 ns (analytical)

~ 120 ns required

= -7.94 kcal/mol (exp) Agreement within 0.25 kcal/mol!

SLIDE 15

protein + ligand* protein:ligand* protein + nothing* protein:nothing*

ΔG*

protein + nothing0 protein:nothing0 protein + ligand0 protein:ligand0

ΔG0 ΔGa

site bulk

ΔGa ΔGc

site bulk

ΔGc

Can use FEP to (de)couple the ligand to the binding site of the protein

ΔGo

site bulk

ΔGo ΔGp

site bulk

ΔGp ΔGa

site bulk

ΔGa “Floating ligand” problem Avoided through definition of a set of restraints Follow a formalism akin to the reaction-coordinate (geometric) route

Alchemical transformations performed bidirectionally using FEP
Bennett acceptance ratio (BAR) estimator
Free-energy contributions due to restraints measured using TI

Gilson, M. K. et al. Biophys. J., 1997, 72, 1047-1069 Bennett, C. H. J. Comp. Phys. , 1976, 22, 245-268

There’s more than one way to…

Most appropriate for buried ligands (no extraction pathway)

SLIDE 16

Φ

Ψ

ϕ

θ

Θ

r

site bulk

Deng, Y.; Roux, B. J. Phys. Chem. B 2009, 113, 2234-2246

The alchemical (FEP) route

SLIDE 17

RMSD PMF PMFs PMFs RMSD

12 ns 8 ns 16 ns

PMF

20 ns 24 ns

decoupling

104 ns

coupling

104 ns

RMSD PMF RMSD

60 ns 48 ns

Comparison of alchemical and geometric routes

SLIDE 18

= -7.8 kcal/mol = -7.7 kcal/mol

Comparison of alchemical and geometric routes

SLIDE 19

Geometrical route Alchemical route RMSD ±0.5 kcal/mol ±0.2 kcal/mol ±0.4 kcal/mol ±0.9 kcal/mol

Low statistical errors
Estimates burdened by systematic error

RMSD ±0.4 kcal/mol ±0.0 kcal/mol ±0.0 kcal/mol ±1.0 kcal/mol alchemy ±0.7 kcal/mol

Hénin, J.; Chipot, C. J. Chem. Phys. 2004, 121, 2904-2914 Rodriguez-Gomez, D. et al. J. Chem. Phys., 2004, 120, 3563-3578 Hahn, A. M.; Then, H. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2009, 80, 031111 Pohorille, A. et al. J. Phys. Chem. B, 2010, 114, 10235-10253

Error analysis

ften very tedious but you

should still do it! (reviewers will often request it anyway!)

SLIDE 20

Geometrical route

Advantages Shortcomings

In principle, applicable to protein:protein dimers
Convergence of RMSD term; Degeneracy
Rigorous, formally correct framework
Access to the statistical error for all terms
Convergence of separation term; ⊥ DoF’s ?
Cumbersome
Limited to interfacial binding sites
Reasonably inexpensive

Alchemical route

Advantages Shortcomings

Convergence of restraint term
In principle, limited to small ligands
Reasonably inexpensive
Rigorous, formally correct framework
Cumbersome
Access to the statistical error for all terms
Convergence of alchemical transformation
Embarrassingly parallelizable

Error analysis

SLIDE 21

Protein-protein binding free energy

Gumbart, Roux, Chipot. Efficient Determination of Protein–Protein Standard Binding Free Energies from First Principles. JCTC 9:3789-3798. 2013. Schreiber & Fersht. JMB, 248:478-486. 1995.

barstar - an inhibitor barnase - a ribonuclease

= -19.0 kcal/mol (exp)

interface is highly solvated

SLIDE 22

Numerous restraints needed

RMSD on barnase backbone RMSD on barnase side chains RMSD on barstar side chains RMSD on barstar backbone

Gumbart, Roux, Chipot. JCTC 9:3789-3798. 2013.

SLIDE 23

Separating the proteins

without side-chain restraints, PMF did not converge even in 400 ns the appropriate choice of restraints is problem dependent!!! PMF took over 50 windows spaced by 0.5 Å and ~200 ns to fully converge

Gumbart, Roux, Chipot. JCTC 9:3789-3798. 2013.

SLIDE 24

Decomposing the PMF

Force decomposition reveals key contributions to the PMF

strong electrostatic component screened by solvent

Gumbart, Roux, Chipot. JCTC 9:3789-3798. 2013.

SLIDE 25

And fourteen separate calculations later…

= -19.0 kcal/mol (exp)

Within 2 kcal/mol!!!

Gumbart, Roux, Chipot. JCTC 9:3789-3798. 2013.

SLIDE 26

Forget everything you just saw: BFEE plugin

Fu et al. BFEE: A user-friendly graphical interface facilitating absolute binding free-energy calculations. J.

Chem. Inf. Model. 2018, 58, 556-560.

A VMD plugin that aids setup and analysis of all the steps to calculate an absolute binding free energy

SLIDE 27

Fu et al. J. Chem. Inf. Model. 2018, 58, 556-560.

Will be widely available in next released version of VMD 1.9.4 To install now, obtain from supplement of published paper